Selmer ranks of quadratic twists of elliptic curves

Details Selmer ranks of quadratic twists of elliptic curves Selmer ranks of quadratic twists of elliptic curves

2011-11-09

arxiv.org/abs/1111.2321v1

Mathematics

Number Theory

Scientific paper

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We first prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. Under the assumption that Gal(K(E[2])/K) = S_3 we also show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on the "parity fraction" alluded to above, of a certain Markov Process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

Affiliated with

Also associated with

No associations

Rating

Selmer ranks of quadratic twists of elliptic curves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Selmer ranks of quadratic twists of elliptic curves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Selmer ranks of quadratic twists of elliptic curves will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-727207